World record formula for nxnxn cubes

[JensRenders]

Not really accurate, but here it is

y=7.5(x-2)^2

where x is the n of the nxnxn cube and y is the estimate average world record

the 7.5 is the value of skill that humans have, the lower this number, the more skill we have, if this number goes down a bit then the time for a 3x3x3 cube would also change a bit and the time of a 7x7x7 cube would change alot more

if the value of 7.5 becomes 0 then we can solve all cubes in no time

I just discovered that 7.5 is what it gives for 3x3x3 haha if you change that to the exact wr 7.64 than thats what the formula gives ofcourse

 

[monkeytherat]

So skill has nothing to do with 2x2 solves??

 

[JensRenders]

2x2 solves can get lower but they are almost zero (and yes, 2.12=0 thats about the accuracy of this formula)

 

[ben1996123]

Single WR estimate:

x	y	WR - y
2	0.43571	0.254287
3	6.63572	-0.97572
4	23.8381	2.9319
5	57.2429	-2.3829
6	112.05	2.76
7	193.46	-0.35
8	306.672
9	456.886
10	649.303
11	889.122

\( y=0.866667x^3-2.29881x^2+1.22738x+0.242857 \)

 

[Sa967St]

Excel gives y = 1.0342e^0.7812x with R² = 0.9525 as the exponential curve of best fit using 3-7 WR averages. Interestingly, the result for 7x7x7 is way under the curve, suggesting it's relatively fast compared to the other WRs.

 

[ben1996123]

Excel gives y = 1.0342e^0.7812x with R² = 0.9525 as the exponential curve of best fit using 3-7 WR averages. Interestingly, the result for 7x7x7 is way under the curve, suggesting it's relatively fast compared to the other WRs.

Wolframalpha gives \( y=2.92575e^{0.606276x} \) with R² = 0.996411 with 3-7 averages. Doesn't really seem that good though

x	y	WR - y
2	9.83653	-7.71653
3	18.0362	-10.3962
4	33.0709	0.4991
5	60.6386	-2.6986
6	111.186	9.244
7	203.87	-3.4
8	373.814
9	685.422
10	1256.78
11	2304.43

 

[Mike Hughey]

Excel gives y = 1.0342e^0.7812x with R² = 0.9525 as the exponential curve of best fit using 3-7 WR averages. Interestingly, the result for 7x7x7 is way under the curve, suggesting it's relatively fast compared to the other WRs.

If you use this formula for 11x11x11, you get 92 minutes!

 

[Sa967St]

If you use this formula for 11x11x11, you get 92 minutes!

Five points isn't enough to accurately extrapolate that far.

 

[JensRenders]

If you use this formula for 11x11x11, you get 92 minutes!

if you use mine, you get 10 minutes!

 

[The Bloody Talon]

Not really accurate, but here it is

y=7.5(x-2)^2

the 7.5 is the value of skill that humans have, the lower this number, the more skill we have, if this number goes down a bit then the time for a 3x3x3 cube would also change a bit and the time of a 7x7x7 cube would change alot more

the value of my skill is approximately 13..

 

[JK]

Oh,that's interesting,how did you find that?

 

[Carrot]

Without too much hack etc I got this:
2.10(x-1)^1.892*floor(x/2)

Kinda close for 1-5:
1 | 0 (spot on yay! )
2 | 2.1 (0.02 off)
3 | 7.79149 (0.15 off)
4 | 33.5709 (spot on)
5 | 57.8558 (0.08 off)
6 | 132.371 (11.94 off)
7 | 186.897 (13.58 off in the other direction than 6x6x6 )

 

[Christopher Mowla]

I think this thread is kind of silly, but to play along, I just came up with a formula. I am not sure what the lowest averages for computer cubes are, so I not sure how values for n greater than 7 compare with human records (computer or by hand).

The Potential time (in seconds) for world record averages of 5 for a cube of size n is:

\( P\left( \text{WR Avg}\text{.}\left( n \right) \right)=\frac{3\text{x3 WR Avg}\text{.}}{\frac{3^{2}}{\ln \left( 3 \right)}\left\lfloor \text{log}_{10}\left( {f}'\left( 3 \right) \right) \right\rfloor }\left\lfloor \text{log}_{10}\left( {f}'\left( n \right) \right) \right\rfloor \frac{n^{2}}{\ln \left( n \right)} \)
, where

\( {f}'\left( n \right)=\text{rate of change of number of positions of a cube of size }n. \)

\( {f}'\left( n \right)=\frac{\left( 8!\times 3^{7} \right)\left( 12!\times 2^{10} \right)24!^{\frac{1}{4}\left( \left( n+1 \right)\left( n-3 \right)+3\cos ^{2}\left( \frac{n\pi }{2} \right) \right)}\ln \left( 24!^{4\left( n-1 \right)}4!^{-24\left( n-2 \right)} \right)}{{}} \)

\( \text{ }8\left( 24\times 12!\times 2^{10} \right)^{\cos ^{2}\left( \frac{n\pi }{2} \right)}4!^{\frac{3}{2}\left( \left( n-1 \right)\left( n-3 \right)+\cos ^{2}\left( \frac{n\pi }{2} \right) \right)} \)
​

Which I first showed in this post.

\( \left\lfloor \text{log}_{10}\left( {f}'\left( n \right) \right) \right\rfloor \) is power of ten of the rate of change for the number of positions for the nxnxn.

So for n = 1 to 1000 (ignore the value for n = 1), if we use the actual derivative (rate of change formula), with Mathematica input:

Spoiler

AccountingForm[(7.64/(3^2/Log[3]*
Floor[Log[
10, ((8! 3^7) (12! 2^10) (Exp[(Log[4008235231]/4 +
Log[6075]/2 + (11/2) Log[2]) ((3 + 1) (3 - 3) +
3 Cos[3 (Pi/2)]^2)] Log[
24!^(4 (3 - 1))/4!^(24 (3 - 2))]))/(8 (24 12! 2^10)^
Cos[3 (Pi/2)]^2 4!^((3/2) ((3 - 1)*(3 - 3) +
Cos[3 (Pi/2)]^2)))]])) (Range[1000]^2/
Log[Range[1000]])*
Floor[Log[
10, ((8! 3^7) (12! 2^10) (Exp[(Log[4008235231]/4 +
Log[6075]/
2 + (11/2) Log[
2]) ((Range[1000] + 1) (Range[1000] - 3) +
3 Cos[Range[1000] (Pi/2)]^2)] Log[
24!^(4 (Range[1000] - 1))/
4!^(24 (Range[1000] - 2))]))/(8 (24 12! 2^10)^
Cos[Range[1000] (Pi/2)]^2 4!^((3/
2) ((Range[1000] - 1)*(Range[1000] - 3) +
Cos[Range[1000] (Pi/2)]^2)))]] // N]

Yields (in seconds, but I included minutes, hours, days, and years)

n     Seconds      Minutes       Hours        Days          Years

2    2.05022    0.034170333    0.00057    2.37294E-05    6.5012E-08       
3    7.64    0.127333333    0.002122    8.84259E-05    2.42263E-07       
4    24.0901    0.401501667    0.006692    0.000278821    7.63892E-07       
5    52.427    0.873783333    0.014563    0.000606794    1.66245E-06       
6    105.288    1.7548    0.029247    0.001218611    3.33866E-06       
7    181.161    3.01935    0.050323    0.002096771    5.74458E-06       
8    299.332    4.988866667    0.083148    0.003464491    9.49176E-06       
9    456.763    7.612716667    0.126879    0.005286609    1.44839E-05       
10    678.896    11.31493333    0.188582    0.007857593    2.15277E-05       
11    956.883    15.94805    0.265801    0.011075035    3.03426E-05       
12    1325.37    22.0895    0.368158    0.015339931    4.20272E-05       
13    1773.2    29.55333333    0.492556    0.020523148    5.62278E-05       
14    2341.76    39.02933333    0.650489    0.027103704    7.42567E-05       
15    3010.87    50.18116667    0.836353    0.034848032    9.54741E-05       
16    3833.91    63.8985    1.064975    0.044373958    0.000121572       
17    4788.17    79.80283333    1.330047    0.055418634    0.000151832       
18    5933.95    98.89916667    1.648319    0.068679977    0.000188164       
19    7236.12    120.602    2.010033    0.083751389    0.000229456       
20    8770.03    146.1671667    2.436119    0.101504977    0.000278096       
21    10498.2    174.97    2.916167    0.121506944    0.000332896       
22    12502.8    208.38    3.473    0.144708333    0.000396461       
23    14730.2    245.5033333    4.091722    0.170488426    0.000467092       
24    17281    288.0166667    4.800278    0.200011574    0.000547977       
25    20099.9    334.9983333    5.583306    0.232637731    0.000637364       
26    23293.6    388.2266667    6.470444    0.269601852    0.000738635       
27    26786.9    446.4483333    7.440806    0.310033565    0.000849407       
28    30708.6    511.81    8.530167    0.355423611    0.000973763       
29    34982.6    583.0433333    9.717389    0.404891204    0.001109291       
30    39743    662.3833333    11.03972    0.459988426    0.001260242       
31    44889.9    748.165    12.46942    0.519559028    0.001423449       
32    50596.2    843.27    14.0545    0.585604167    0.001604395       
33    56736.8    945.6133333    15.76022    0.656675926    0.001799112       
34    63473.7    1057.895    17.63158    0.734649306    0.002012738       
35    70722.9    1178.715    19.64525    0.818552083    0.002242608       
36    78634.4    1310.573333    21.84289    0.91012037    0.00249348       
37    87097.2    1451.62    24.19367    1.008069444    0.002761834       
38    96307.7    1605.128333    26.75214    1.114672454    0.003053897       
39    106108    1768.466667    29.47444    1.228101852    0.003364663       
40    116747    1945.783333    32.42972    1.351238426    0.003702023       
41    128013    2133.55    35.55917    1.481631944    0.004059266       
42    140196    2336.6    38.94333    1.622638889    0.004445586       
43    153105    2551.75    42.52917    1.772048611    0.004854928       
44    166969    2782.816667    46.38028    1.932511574    0.005294552       
45    181623    3027.05    50.45083    2.102118056    0.005759228       
46    197335    3288.916667    54.81528    2.283969907    0.006257452       
47    213877    3564.616667    59.41028    2.475428241    0.006781995       
48    231588    3859.8    64.33    2.680416667    0.007343607       
49    250197    4169.95    69.49917    2.895798611    0.007933695       
50    270037    4500.616667    75.01028    3.125428241    0.008562817       
51    290842    4847.366667    80.78944    3.366226852    0.009222539       
52    312969    5216.15    86.93583    3.622326389    0.009924182       
53    336162    5602.7    93.37833    3.890763889    0.010659627       
54    360772    6012.866667    100.2144    4.175601852    0.011440005       
55    386523    6442.05    107.3675    4.473645833    0.012256564       
56    413755    6895.916667    114.9319    4.788831019    0.013120085       
57    442204    7370.066667    122.8344    5.118101852    0.014022197       
58    472268    7871.133333    131.1856    5.466064815    0.01497552       
59    503629    8393.816667    139.8969    5.829039352    0.015969971       
60    536670    8944.5    149.075    6.211458333    0.017017694       
61    571089    9518.15    158.6358    6.609826389    0.018109113       
62    607331    10122.18333    168.7031    7.029293981    0.01925834       
63    645037    10750.61667    179.1769    7.465706019    0.020453989       
64    684631    11410.51667    190.1753    7.923969907    0.021709507       
65    725775    12096.25    201.6042    8.400173611    0.023014174       
66    768961    12816.01667    213.6003    8.900011574    0.024383593       
67    813785    13563.08333    226.0514    9.41880787    0.025804953       
68    860720    14345.33333    239.0889    9.962037037    0.027293252       
69    909382    15156.36667    252.6061    10.52525463    0.028836314       
70    960318    16005.3    266.755    11.11479167    0.030451484       
71    1013076    16884.6    281.41    11.72541667    0.032124429       
72    1068176    17802.93333    296.7156    12.36314815    0.033871639       
73    1125192    18753.2    312.5533    13.02305556    0.035679604       
74    1184782    19746.36667    329.1061    13.71275463    0.037569191       
75    1246273    20771.21667    346.1869    14.42445602    0.039519058       
76    1310464    21841.06667    364.0178    15.16740741    0.041554541       
77    1376710    22945.16667    382.4194    15.93414352    0.043655188       
78    1445727    24095.45    401.5908    16.73295139    0.045843702       
79    1516896    25281.6    421.36    17.55666667    0.048100457       
80    1591031    26517.18333    441.9531    18.41471065    0.050451262       
81    1667420    27790.33333    463.1722    19.29884259    0.052873541       
82    1746845    29114.08333    485.2347    20.21811343    0.055392092       
83    1828696    30478.26667    507.9711    21.16546296    0.05798757       
84    1913650    31894.16667    531.5694    22.14872685    0.060681443       
85    2001209    33353.48333    555.8914    23.1621412    0.063457921       
86    2092009    34866.81667    581.1136    24.21306713    0.06633717       
87    2185378    36422.96667    607.0494    25.29372685    0.069297882       
88    2282276    38037.93333    633.9656    26.41523148    0.072370497       
89    2381854    39697.56667    661.6261    27.56775463    0.075528095       
90    2485031    41417.18333    690.2864    28.76193287    0.078799816       
91    2591079    43184.65    719.7442    29.98934028    0.082162576       
92    2700875    45014.58333    750.2431    31.26012731    0.085644184       
93    2813577    46892.95    781.5492    32.56454861    0.089217941       
94    2930174    48836.23333    813.9372    33.91405093    0.092915208       
95    3049880    50831.33333    847.1889    35.29953704    0.09671106       
96    3173546    52892.43333    881.5406    36.73085648    0.100632484       
97    3300440    55007.33333    916.7889    38.19953704    0.104656266       
98    3431631    57193.85    953.2308    39.71795139    0.108816305       
99    3565988    59433.13333    990.5522    41.27300926    0.113076738       
100    3704803    61746.71667    1029.112    42.87966435    0.117478532       
101    3846999    64116.65    1068.611    44.52545139    0.121987538       
102    3993818    66563.63333    1109.394    46.22474537    0.126643138       
103    4144043    69067.38333    1151.123    47.96346065    0.131406742       
104    4299055    71650.91667    1194.182    49.75758102    0.13632214       
105    4457701    74295.01667    1238.25    51.59376157    0.141352771       
106    4621198    77019.96667    1283.666    53.48608796    0.146537227       
107    4788563    79809.38333    1330.156    55.42318287    0.151844337       
108    4960951    82682.51667    1378.042    57.41841435    0.157310724       
109    5137229    85620.48333    1427.008    59.45866898    0.162900463       
110    5318815    88646.91667    1477.449    61.5603588    0.168658517       
111    5504309    91738.48333    1528.975    63.70728009    0.174540493       
112    5695285    94921.41667    1582.024    65.91765046    0.180596303       
113    5890419    98173.65    1636.228    68.17614583    0.186783961       
114    6091219    101520.3167    1692.005    70.50021991    0.193151287       
115    6296190    104936.5    1748.942    72.87256944    0.199650875       
116    6507007    108450.1167    1807.502    75.31258102    0.206335838       
117    6722257    112037.6167    1867.294    77.80390046    0.213161371       
118    6943540    115725.6667    1928.761    80.3650463    0.220178209       
119    7169398    119489.9667    1991.499    82.97914352    0.227340119       
120    7401214    123353.5667    2055.893    85.66219907    0.234690956       
121    7638011    127300.1833    2121.67    88.40290509    0.24219974       
122    7880951    131349.1833    2189.153    91.21471065    0.249903317       
123    8128887    135481.45    2258.024    94.08434028    0.257765316       
124    8383292    139721.5333    2328.692    97.02884259    0.265832445       
125    8642702    144045.0333    2400.751    100.0312731    0.274058283       
126    8908916    148481.9333    2474.699    103.1124537    0.282499873       
127    9180138    153002.3    2550.038    106.2515972    0.291100266       
128    9458362    157639.3667    2627.323    109.4717824    0.299922692       
129    9742041    162367.35    2706.123    112.7551042    0.308918094       
130    10032622    167210.3667    2786.839    116.1183102    0.318132357       
131    10328813    172146.8833    2869.115    119.5464468    0.327524512       
132    10632258    177204.3    2953.405    123.0585417    0.337146689       
133    10941313    182355.2167    3039.254    126.6355671    0.346946759       
134    11257985    187633.0833    3127.218    130.3007523    0.356988363       
135    11580260    193004.3333    3216.739    134.030787    0.367207636       
136    11910360    198506    3308.433    137.8513889    0.377675038       
137    12246555    204109.25    3401.821    141.7425347    0.388335712       
138    12590449    209840.8167    3497.347    145.7227894    0.399240519       
139    12940600    215676.6667    3594.611    149.775463    0.410343734       
140    13298831    221647.1833    3694.12    153.9216551    0.421703165       
141    13663309    227721.8167    3795.364    158.1401505    0.433260686       
142    14036259    233937.65    3898.961    162.4567014    0.445086853       
143    14415622    240260.3667    4004.339    166.8474769    0.457116375       
144    14803493    246724.8833    4112.081    171.3367245    0.469415684       
145    15197942    253299.0333    4221.651    175.9021065    0.481923579       
146    15601305    260021.75    4333.696    180.5706597    0.494714136       
147    16011416    266856.9333    4447.616    185.3173148    0.507718671       
148    16430474    273841.2333    4564.021    190.1675231    0.521006913       
149    16856448    280940.8    4682.347    195.0977778    0.53451446       
150    17291789    288196.4833    4803.275    200.1364468    0.548319032       
151    17734220    295570.3333    4926.172    205.2571759    0.562348427       
152    18186046    303100.7667    5051.679    210.4866435    0.576675736       
153    18645135    310752.25    5179.204    215.8001736    0.591233352       
154    19114053    318567.55    5309.459    221.2274653    0.606102645       
155    19590412    326506.8667    5441.781    226.7408796    0.621207889       
156    20076625    334610.4167    5576.84    232.3683449    0.636625602       
157    20570454    342840.9    5714.015    238.0839583    0.652284817       
158    21074587    351243.1167    5854.052    243.918831    0.66827077       
159    21586517    359775.2833    5996.255    249.8439468    0.684503964       
160    22108771    368479.5167    6141.325    255.8885532    0.701064529       
161    22639228    377320.4667    6288.674    262.0281019    0.717885211       
162    23180021    386333.6833    6438.895    268.2872801    0.735033644       
163    23729202    395486.7    6591.445    274.6435417    0.752448059       
164    24289187    404819.7833    6746.996    281.1248495    0.770205067       
165    24857754    414295.9    6904.932    287.7054861    0.788234209       
166    25437129    423952.15    7065.869    294.4112153    0.806606069       
167    26025278    433754.6333    7229.244    301.2184954    0.825256152       
168    26624963    443749.3833    7395.823    308.159294    0.844272038       
169    27233128    453885.4667    7564.758    315.1982407    0.863556824       
170    27853080    464218    7736.967    322.3736111    0.883215373       
171    28481949    474699.15    7911.653    329.6521875    0.903156678       
172    29122608    485376.8    8089.613    337.0672222    0.923471842       
173    29772633    496210.55    8270.176    344.5906597    0.944083999       
174    30434443    507240.7167    8454.012    352.2504977    0.965069857       
175    31105816    518430.2667    8640.504    360.0210185    0.986358955       
176    31789754    529829.2333    8830.487    367.9369676    1.008046487       
177    32482929    541382.15    9023.036    375.9598264    1.030026922       
178    33188929    553148.8167    9219.147    384.1311227    1.052414035       
179    33904631    565077.1833    9417.953    392.4147106    1.075108796       
180    34633150    577219.1667    9620.319    400.8466435    1.098209982       
181    35371849    589530.8167    9825.514    409.3964005    1.121633974       
182    36123631    602060.5167    10034.34    418.097581    1.145472825       
183    36885519    614758.65    10245.98    426.9157292    1.169632135       
184    37660753    627679.2167    10461.32    435.8883449    1.194214644       
185    38446584    640776.4    10679.61    444.9836111    1.219133181       
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419    877146739    14619112.32    243651.9    10152.16133    27.81414063       
420    885209713    14753495.22    245891.6    10245.48279    28.06981586       
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426    934721210    15578686.83    259644.8    10818.53252    29.63981513       
427    943165528    15719425.47    261990.4    10916.26769    29.9075827       
428    951671962    15861199.37    264353.3    11014.72178    30.17731995       
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922    18211750308    303529171.8    5058820    210784.1471    577.4908139       
923    18287993443    304799890.7    5079998    211666.5908    579.9084679       
924    18364496735    306074945.6    5101249    212552.0455    582.3343714       
925    18441210731    307353512.2    5122559    213439.939    584.7669562       
926    18518191498    308636525    5143942    214330.9201    587.2080003       
927    18595378248    309922970.8    5165383    215224.2853    589.6555761       
928    18672838433    311213973.9    5186900    216120.8152    592.1118225       
929    18750505442    312508424    5208474    217019.7389    594.5746272       
930    18828441379    313807356.3    5230123    217921.7752    597.0459595       
931    18906584963    315109749.4    5251829    218826.2148    599.5238763       
932    18985004185    316416736.4    5273612    219733.8447    602.0105335       
933    19063631902    317727198.4    5295453    220643.8878    604.5038021       
934    19142530708    319042178.5    5317370    221557.0684    607.0056668       
935    19221638834    320360647.2    5339344    222472.6717    609.514169       
936    19301024810    321683746.8    5361396    223391.4909    612.0314818       
937    19380615256    323010254.3    5383504    224312.6766    614.5552783       
938    19460490361    324341506    5405692    225237.157    617.0881012       
939    19540570764    325676179.4    5427936    226164.0135    619.6274342       
940    19620931239    327015520.7    5450259    227094.1116    622.1756481       
941    19701497819    328358297    5472638    228026.5951    624.7303976       
942    19782351328    329705855.5    5495098    228962.3996    627.2942456       
943    19863411775    331056862.9    5517614    229900.5992    629.8646555       
944    19944754521    332412575.4    5540210    230842.0662    632.444017       
945    20026310805    333771846.8    5562864    231786.0047    635.0301498       
946    20108144717    335135745.3    5585596    232733.1564    637.6250862       
947    20190193001    336503216.7    5608387    233682.7894    640.2268202       
948    20272525822    337875430.4    5631257    234635.7155    642.8375768       
949    20355073874    339251231.2    5654187    235591.1328    645.4551584       
950    20437901750    340631695.8    5677195    236549.7888    648.0816131       
951    20520945695    342015761.6    5700263    237510.9455    650.7149193       
952    20604276422    343404607    5723410    238475.4216    653.3573193       
953    20687824079    344797068    5746618    239442.4083    656.0065981       
954    20771653766    346194229.4    5769904    240412.6593    658.6648201       
955    20855707126    347595118.8    5793252    241385.4991    661.3301346       
956    20940037717    349000628.6    5816677    242361.5477    664.0042401       
957    21024592848    350409880.8    5840165    243340.195    666.6854658       
958    21109438156    351823969.3    5863733    244322.2009    669.3758928       
959    21194497019    353241617    5887360    245306.6785    672.0730917       
960    21279847181    354664119.7    5911069    246294.5276    674.7795276       
961    21365417692    356090294.9    5934838    247284.927    677.4929507       
962    21451274663    357521244.4    5958687    248278.6419    680.2154574       
963    21537358825    358955980.4    5982600    249274.9864    682.9451682       
964    21623724566    360395409.4    6006590    250274.5899    685.6838079       
965    21710324388    361838739.8    6030646    251276.9026    688.4298702       
966    21797206893    363286781.6    6054780    252282.4872    691.1848964       
967    21884312315    364738538.6    6078976    253290.6518    693.9469912       
968    21971713584    366195226.4    6103254    254302.2406    696.7184673       
969    22059338647    367655644.1    6127594    255316.4195    699.4970398       
970    22147254634    369120910.6    6152015    256333.9657    702.2848375       
971    22235401355    370590022.6    6176500    257354.1823    705.0799516       
972    22323840134    372064002.2    6201067    258377.7793    707.8843269       
973    22412504438    373541740.6    6225696    259403.9866    710.6958536       
974    22501461910    375024365.2    6250406    260433.5869    713.5166765       
975    22590651898    376510865    6275181    261465.8784    716.3448725       
976    22680136192    378002269.9    6300038    262501.5763    719.1824008       
977    22769847747    379497462.5    6324958    263539.9045    722.0271356       
978    22859854725    380997578.8    6349960    264581.6519    724.8812381       
979    22950096004    382501600.1    6375027    265626.1112    727.7427703       
980    23040633847    384010564.1    6400176    266674.0029    730.6137065       
981    23131400695    385523344.9    6425389    267724.5451    733.4919043       
982    23222465226    387041087.1    6450685    268778.5327    736.3795417       
983    23313765853    388562764.2    6476046    269835.2529    739.2746656       
984    23405365307    390089421.8    6501490    270895.4318    742.1792652       
985    23497195517    391619925.3    6526999    271958.2814    745.091182       
986    23589325678    393155428    6552590    273024.6028    748.0126103       
987    23681693735    394694895.6    6578248    274093.6775    750.9415822       
988    23774362893    396239381.6    6603990    275166.2372    753.8801019       
989    23867264563    397787742.7    6629796    276241.488    756.8259945       
990    23960468460    399341141    6655686    277320.2368    759.7814707       
991    24053912061    400898534.4    6681642    278401.76    762.7445479       
992    24147659042    402460984    6707683    279486.7945    765.7172451       
993    24241640298    404027338.3    6733789    280574.5405    768.6973712       
994    24335926067    405598767.8    6759979    281665.811    771.6871533       
995    24430453352    407174222.5    6786237    282759.8768    774.6845939       
996    24525286307    408754771.8    6812580    283857.4804    777.6917271       
997    24620355305    410339255.1    6838988    284957.816    780.7063453       
998    24715737515    411928958.6    6865483    286061.7768    783.7308953       
999    24811356666    413522611.1    6892044    287168.4799    786.7629587       
1000    24907283784    415121396.4    6918690    288278.7475    789.8047877

For \( \left\lfloor \log _{10}\left( {f}'\left( n \right) \right) \right\rfloor \), I made the approximation formula (which is accurate up to n = 31).

\( \left\lfloor \log _{10}\left( {f}'\left( n \right) \right) \right\rfloor \approx \frac{1}{16}\left( 62n^{2}-56n-4\cos \left( \frac{n\pi }{2} \right)+21\cos \left( n\pi \right)-33 \right) \)

which I first showed in this post.

By using this approximation, we can have a formula at wolframalpha.

\( \frac{3\text{x3 WR Avg}\text{.}}{\frac{3^{2}}{\ln \left( 3 \right)}\left( 21 \right)}\frac{n^{2}}{\ln \left( n \right)}\frac{1}{16}\left( 62n^{2}-56n-4\cos \left( \frac{n\pi }{2} \right)+21\cos \left( n\pi \right)-33 \right) \)

Note that \( \left\lfloor \log _{10}\left( {f}'\left( 3 \right) \right) \right\rfloor =21 \) with the approximation formula as well as the actual derivative formula (that's where the 21 comes from).

To get values for a certain n, just change the 2 (located on the far right of the input bar) to a different number in the input bar.

 

[ben1996123]

x	y
2	0.669408
3	5.59801
4	26.6197
5	54.5381
6	154.179
7	191.958
8	-804.047
9	-5651.51

Pretty much exact for 2-7, but lol8,9.

\( y=-0.510403x^6+11.5106x^5-102.079x^4+453.215x^3-1047.51x^2+1177.21x-491.84 \)

 

[RNewms27]

I'm curious to see a BLD formula

 

[Godmil]

y=7.5(x-2)^2

\( P\left( \text{WR Avg}\text{.}\left( n \right) \right)=\frac{3\text{x3 WR Avg}\text{.}}{\frac{3^{2}}{\ln \left( 3 \right)}\left\lfloor \text{log}_{10}\left( {f}'\left( 3 \right) \right) \right\rfloor }\left\lfloor \text{log}_{10}\left( {f}'\left( n \right) \right) \right\rfloor \frac{n^{2}}{\ln \left( n \right)} \)
, where

\( {f}'\left( n \right)=\text{rate of change of number of positions of a cube of size }n. \)

\( {f}'\left( n \right)=\frac{\left( 8!\times 3^{7} \right)\left( 12!\times 2^{10} \right)24!^{\frac{1}{4}\left( \left( n+1 \right)\left( n-3 \right)+3\cos ^{2}\left( \frac{n\pi }{2} \right) \right)}\ln \left( 24!^{4\left( n-1 \right)}4!^{-24\left( n-2 \right)} \right)}{{}} \)

\( \text{ }8\left( 24\times 12!\times 2^{10} \right)^{\cos ^{2}\left( \frac{n\pi }{2} \right)}4!^{\frac{3}{2}\left( \left( n-1 \right)\left( n-3 \right)+\cos ^{2}\left( \frac{n\pi }{2} \right) \right)} \)
​

You know what, I'm going to give JensRenders the point for the elegance of his formula.